How to Solve Quadratic Equations
There are three standard ways to solve a quadratic equation, and each has a situation where it is fastest. This guide explains all three with worked examples, then shows how to choose between them.
Step zero: write it as ax² + bx + c = 0
Before any method, move every term to one side so the equation equals zero. For instance, x² + 2x = 8 becomes x² + 2x − 8 = 0. Then read off a, b and c, with their signs.
Method 1: factoring
Find two numbers that multiply to c and add to b, write the brackets, and set each to zero. It is the quickest method when the numbers are small and the quadratic factors.
Method 2: completing the square
Rewrite x² + bx as (x + b/2)² − (b/2)². This always works and gives the vertex form of the parabola, which helps with graphing. It is especially good when b is even.
Method 3: the quadratic formula
Use x = (−b ± √(b² − 4ac)) / 2a. It works for every quadratic, including ones that do not factor. Practise it with the quadratic equation solver.
Which method should you pick?
| Situation | Best method |
|---|---|
| Small whole-number roots, a = 1 | Factoring |
| Need the vertex, or b is even | Completing the square |
| Messy numbers, or nothing factors | Quadratic formula |
| No x term, such as x² − 16 = 0 | Square roots directly |
If the factoring step trips you up, try the factoring calculator, then check the shape on the graphing solver.
Worked examples
Example 1: Factoring: x² + 2x − 8 = 0
- Find two numbers with product −8 and sum 2: 4 and −2.
- Write (x + 4)(x − 2) = 0.
- Set each bracket to zero.
Final answer: x = −4 or x = 2
Example 2: Completing the square: x² + 6x + 2 = 0
- Move the constant: x² + 6x = −2.
- Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7.
- Write as (x + 3)² = 7, then x + 3 = ±√7.
Final answer: x = −3 ± √7 (≈ −0.354 or −5.646)
Example 3: Formula: 3x² − 4x − 1 = 0
- a = 3, b = −4, c = −1.
- D = 16 − 4(3)(−1) = 16 + 12 = 28, and √28 = 2√7.
- x = (4 ± 2√7) / 6 = (2 ± √7) / 3.
Final answer: x = (2 ± √7) / 3 (≈ 1.549 or −0.215)
Common mistakes to avoid
- Trying to factor before the equation equals zero.
- Choosing factor pairs with the right product but the wrong sum.
- Forgetting to add the same number to both sides when completing the square.
- Losing the ± when taking a square root.
- Mishandling signs for b in the formula.
Frequently asked questions
Which method is best for beginners?
Start with factoring for simple cases and learn the formula as your safety net, since it never fails.
Can a quadratic have no real solutions?
Yes, when the discriminant is negative. The solutions are then complex numbers.
Why is there ± in the formula?
Because a square root has a positive and a negative value, and both give valid solutions.
How do I check my roots?
Substitute each into the original equation. Both should give 0.
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